Dual module
In mathematics, the dual module of a left (respectively right) module M over a ring R is the set of left (respectively right) R-module homomorphisms from M to R with the pointwise right (respectively left) module structure. The dual module is typically denoted M∗ or HomR(M, R). If the base ring R is a field, then a dual module is a dual vector space. Every module has a canonical homomorphism to the dual of its dual (called the double dual). A reflexive module is one for which the canonical homomorphism is an isomorphism. A torsionless module is one for which the canonical homomorphism is injective. Example: If G = Spec ( A ) {\displaystyle G=\operatorname {Spec} (A)} is a finite commutative group scheme represented by a Hopf algebra A over a commutative ring R, then the Cartier dual G D {\displaystyle G^{D}} is the Spec of the dual R-module of A.
In mathematics, the dual module of a left (respectively right) module M over a ring R is the set of left (respectively right) R-module homomorphisms from M to R with the pointwise right (respectively left) module structure.[1][2] The dual module is typically denoted M∗ or HomR(M, R).
If the base ring R is a field, then a dual module is a dual vector space.
Every module has a canonical homomorphism to the dual of its dual (called the double dual). A reflexive module is one for which the canonical homomorphism is an isomorphism. A torsionless module is one for which the canonical homomorphism is injective.
Example: If is a finite commutative group scheme represented by a Hopf algebra A over a commutative ring R, then the Cartier dual is the Spec of the dual R-module of A.
See also
[edit]- Dual sheaf of a sheaf of modules
References
[edit]- ^ Nicolas Bourbaki (1974). Algebra I. Springer. ISBN 9783540193739.
- ^ Serge Lang (2002). Algebra. Springer. ISBN 978-0387953854.
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